joe N says: > One can say "just accept it, because it makes other things work out nicely". That's the "formal" approach.
PT III says >Not to some of us. The formal approach is to prove it.
Joe N: > Or one can attempt to understand it another way that grounds it in the everyday.
PT III: >Or one can prove it (where "it" could be the more general -(a/b) = a/(-b) for all elements in the set even if the set contains no identity element).
You no doubt "help" your 4th graders this way. Well, that helps weed 'em out I bet.
Now, here's a better story -- this is not a 4th grade lesson but a meta-level observation on what and how we teach numbers.
Negative numbers, and everything that goes with them (sign rules) are useful in some contexts and useless or absurd in other contexts. That's part of learning mathematics -- how to map the concepts from the everyday contexts (e.g., accounting) to the symbols and rules we learn in arithmetic and algebra etc. That "mapping" is crucial to making sense out of the whole enterprise. It is also given short shrift by people like PT III and R.H.
The fact is all the various number systems and other mathematical structures, in so far as they are useful, find there use in some contexts and not others. (This is part of the reason why Devlin's view that students should be taught the integers as a "restriction" from real numbers is so patently absurd.)
Here's a nice quote from "Negative Math" (Matinez), he is himself quoting Henri Beyle, recalling his youth in math class:
"I remember distinctly that, when I spoke of my difficult about *minus times minus* to one of the experts, he laughed in my face; they were all more or less like Paul-Emile Teisseire, and used to learn by rote. I often watched them say at the blackboard, at the end of their demonstrations: 'It is therefore evident...' Nothing is less evident to you, I would think."
[i.e., he thinks they are just parroting without understanding -- Joe N]
"But the things in question were evident to me..."
[i.e., the absurdities he could easily find in inappropriate mappings were quite obvious and forceful to his common sense -- Joe N]
How much better it would be if we taught children, when going from whole numbers to integers (i.e., negative numbers), that this is a *new* game, that has some similarities to the old game, but also some differences. For some problems the old game is perfectly suited, in others the new game will help, and you must also learn to see which is which.
But there is a deep seated prejudice against that sort of teaching. We *want* mathematics to be some beautiful seamless edifice, and so we pretend that it is, even though some children can see the emperor has no clothes.